Finding the graph formula with know points and other equation.

[Pharrahnox]

I have an equation for determining the acceleration of an object being propelled by a constant power source, that is affected by air resistance:

a = \frac{P}{mv}-\frac{C_DpAv²}{2m}

Since F = \frac{P}{v}

I am trying to graph this as a velocity-time graph, however, I don't know how to do it. There is no time variable that I can replace with x, and the y-value (velocity) is mixed into the equation already.

I remember the equation given to me for a similar sort of thing, without air resitance, but instead just a constant friction force, that was something like this:

y = k(1-e^-ax)

Where k is a constant, which is the maximum speed, and a is another constant which represents the force of air resistance.

The maximum speed in this case is \sqrt[3]{\frac{2P}{C_DpA}}, so the equation would be something like:

y = \sqrt[3]{\frac{2P}{C_DpA}}(1-e^-ax)

But that's as far as I've gotten. By the use of iteration, I have determined the velocity at several different times, here's a few, just in case it helps:

(0,0) (25,83.4762) (50,118.1195) (75,126.1601) (100,127.7624) (125,128.0709)

This is for variables of values: p = 0.001, A = 1900.933, m = 1900.933, P = 1*10⁶
and C_D = 0.5

Any help would be greatly appreciated, and if you need any more information, just let me know.EDIT: the equations don't seem to be formatting correctly, so I'll redo them down here:

a = P/v - (Cd*p*A*v^2) /2
F = P/v
max speed = ( (2*P) / (Cd*p*A) )^1/3
y = ( (2*P) / (Cd*p*A) )^1/3 * (1 - e^-ax)

 

[gopher_p]

Given $a=\frac{dv}{dt}$, what you have is a separable ordinary differential equation \frac{dv}{dt}=\frac{\alpha}{v}-\beta v^2 where $\alpha=\frac{P}{m}$ and $\beta=\frac{C_DpA}{2m}$ have been substituted to make the equation (slightly) more manageable. Hopefully that's enough to get you going.

I'm not sure where you got your force equation. Are you not using $F=ma$? Furthermore, I don't know that it's even useful here.

I reckon it's doubtful that the equation that works in a constant friction situation applies here. So you're likely barking up the wrong tree there.

 

[Pharrahnox]

I have never done differential equations before, and I tried to just find the anti-derivative, but my teacher said that I had done it incorrectly, as I differentiated dv instead of dt, or the other way around. Anyway, here is what I got:

v = (Pt/m)*ln(|v|)-(C_DpAv³t)/(6m)+c

Unfortunately from there, if it is correct, I don't know where to go - how to get v by itself without v on the other side.

What do I do from here, or from the start if that isn't correct?

Thanks for your response.

 

[Stephen Tashi]

the equations don't seem to be formatting correctly

I think this is due to your using the SUB and SUP tags instead of the usual LaTex notation.

a = \frac{P}{v} - \frac{ (C_d p A v^2) }{2}

 

[Pharrahnox]

Oh ok, I'll give it a go:

a = (\frac{Pt}{m})ln(|v|)-\frac{C_{D}pAv^{3}t}{6m}

Seems to work, thanks.

 

[Pharrahnox]

Does anyone have any advice on what I should do to get time into the equation, without velocity on both sides? I don't even know what to type into google to find information on it. Any information would be much appreciated.